Fourier multipliers on compact groups

In a recent paperIan Inglis gives some sufficient conditions for a function on a totally disconnected compact Abelian group to be anL^p Fourier multiplier. His proof depends on an interpolation theorem ofE. M. Stein. In this note we prove a generalization ofInglis' theorem. Our result is deduced from a factorization theorem, the proof of which is elementary, and a standard multiplier theorem.This work was done while the second-named author held a visiting appointment at the University of Washington. He wishes to thank ProfessorsE. Hewitt andR. R. Phelps for making the visit possible.